P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 185

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆,使共同半径尽可能大。

子题n = 185
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0669427168509965103782990720155, 0.0669427168509965103782990720165]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆,圆心在 (1, 1),两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆:一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内;两两内部不重叠,相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示:起步布局

2y0
0x2
已验证构造r = 0.066942716

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构:圆彼此顶住、顶住边界,常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明;其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型,大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 185 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路,审核采纳后可获得证明分。
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆,圆心在 (1, 1),所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位,直接写成小数,例如 "0.5",最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "1",
      "0.068950998"
    ],
    [
      "0.866114566",
      "0.076598362"
    ],
    [
      "1.133885434",
      "0.076598362"
    ],
    [
      "0.68571215",
      "0.121467678"
    ],
    [
      "1.31428785",
      "0.121467678"
    ],
    [
      "0.563210975",
      "0.175493334"
    ],
    [
      "1.436789025",
      "0.175493334"
    ],
    [
      "0.799171849",
      "0.192546549"
    ],
    [
      "0.933057283",
      "0.192546549"
    ],
    [
      "1.066942717",
      "0.192546549"
    ],
    [
      "1.200828151",
      "0.192546549"
    ],
    [
      "0.449703156",
      "0.246495337"
    ],
    [
      "1.550296844",
      "0.246495337"
    ],
    [
      "0.59839828",
      "0.304672137"
    ],
    [
      "1.40160172",
      "0.304672137"
    ],
    [
      "0.732229133",
      "0.308494736"
    ],
    [
      "0.866114566",
      "0.308494736"
    ],
    [
      "1",
      "0.308494736"
    ],
    [
      "1.133885434",
      "0.308494736"
    ],
    [
      "1.267770867",
      "0.308494736"
    ],
    [
      "0.406271757",
      "0.373140601"
    ],
    [
      "1.593728243",
      "0.373140601"
    ],
    [
      "0.27745694",
      "0.409637891"
    ],
    [
      "1.72254306",
      "0.409637891"
    ],
    [
      "0.531455563",
      "0.420620323"
    ],
    [
      "1.468544437",
      "0.420620323"
    ],
    [
      "0.665286416",
      "0.424442923"
    ],
    [
      "0.799171849",
      "0.424442923"
    ],
    [
      "0.933057283",
      "0.424442923"
    ],
    [
      "1.066942717",
      "0.424442923"
    ],
    [
      "1.200828151",
      "0.424442923"
    ],
    [
      "1.334713584",
      "0.424442923"
    ],
    [
      "0.194795964",
      "0.532485703"
    ],
    [
      "0.330689679",
      "0.532485703"
    ],
    [
      "1.669310321",
      "0.532485703"
    ],
    [
      "1.805204036",
      "0.532485703"
    ],
    [
      "0.464512846",
      "0.53656851"
    ],
    [
      "1.535487154",
      "0.53656851"
    ],
    [
      "0.598343699",
      "0.54039111"
    ],
    [
      "0.732229133",
      "0.54039111"
    ],
    [
      "0.866114566",
      "0.54039111"
    ],
    [
      "1",
      "0.54039111"
    ],
    [
      "1.133885434",
      "0.54039111"
    ],
    [
      "1.267770867",
      "0.54039111"
    ],
    [
      "1.401656301",
      "0.54039111"
    ],
    [
      "0.263746962",
      "0.648433889"
    ],
    [
      "1.736253038",
      "0.648433889"
    ],
    [
      "0.39757013",
      "0.652516697"
    ],
    [
      "1.60242987",
      "0.652516697"
    ],
    [
      "0.531400982",
      "0.656339296"
    ],
    [
      "0.665286416",
      "0.656339296"
    ],
    [
      "0.799171849",
      "0.656339296"
    ],
    [
      "0.933057283",
      "0.656339296"
    ],
    [
      "1.066942717",
      "0.656339296"
    ],
    [
      "1.200828151",
      "0.656339296"
    ],
    [
      "1.334713584",
      "0.656339296"
    ],
    [
      "1.468599018",
      "0.656339296"
    ],
    [
      "0.130502423",
      "0.661518307"
    ],
    [
      "1.869497577",
      "0.661518307"
    ],
    [
      "0.330627413",
      "0.768464884"
    ],
    [
      "1.669372587",
      "0.768464884"
    ],
    [
      "0.464458265",
      "0.772287483"
    ],
    [
      "0.598343699",
      "0.772287483"
    ],
    [
      "0.732229133",
      "0.772287483"
    ],
    [
      "0.866114566",
      "0.772287483"
    ],
    [
      "1",
      "0.772287483"
    ],
    [
      "1.133885434",
      "0.772287483"
    ],
    [
      "1.267770867",
      "0.772287483"
    ],
    [
      "1.401656301",
      "0.772287483"
    ],
    [
      "1.535541735",
      "0.772287483"
    ],
    [
      "0.197060037",
      "0.777687981"
    ],
    [
      "1.802939963",
      "0.777687981"
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      "0.397448606",
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    ],
    [
      "1.932757182",
      "0.976337079"
    ],
    [
      "0.464458265",
      "1.004183857"
    ],
    [
      "0.598343699",
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    ],
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    ],
    [
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    ],
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      "1.535541735",
      "1.004183857"
    ],
    [
      "0.330687783",
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    ],
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    [
      "0.196802841",
      "1.010093707"
    ],
    [
      "1.803197159",
      "1.010093707"
    ],
    [
      "0.073458745",
      "1.110078141"
    ],
    [
      "1.926541255",
      "1.110078141"
    ],
    [
      "0.531400982",
      "1.120132043"
    ],
    [
      "0.665286416",
      "1.120132043"
    ],
    [
      "0.799171849",
      "1.120132043"
    ],
    [
      "0.933057283",
      "1.120132043"
    ],
    [
      "1.066942717",
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    ],
    [
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    [
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    [
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    ],
    [
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    [
      "1.671778813",
      "1.243490007"
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    [
      "0.132020135",
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    [
      "1.867979865",
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    [
      "0.665286416",
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    [
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      "1.604871893"
    ],
    [
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      "1.608279513"
    ],
    [
      "1.578258598",
      "1.608279513"
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    [
      "0.545005875",
      "1.661628774"
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    [
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      "1.738264496"
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    [
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      "1.741621662"
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      "1.81502344"
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    [
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    [
      "1.066942717",
      "1.815821164"
    ],
    [
      "0.667084985",
      "1.871644128"
    ],
    [
      "1.332915015",
      "1.871644128"
    ],
    [
      "0.850145924",
      "1.920944976"
    ],
    [
      "1.149854076",
      "1.920944976"
    ],
    [
      "1",
      "1.93299034"
    ]
  ],
  "radius": "0.066942716"
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看+

子题参数

{
  "n": 185
}

起步布局答案

{
  "centers": [
    [
      "1",
      "0.068950998"
    ],
    [
      "0.866114566",
      "0.076598362"
    ],
    [
      "1.133885434",
      "0.076598362"
    ],
    [
      "0.68571215",
      "0.121467678"
    ],
    [
      "1.31428785",
      "0.121467678"
    ],
    [
      "0.563210975",
      "0.175493334"
    ],
    [
      "1.436789025",
      "0.175493334"
    ],
    [
      "0.799171849",
      "0.192546549"
    ],
    [
      "0.933057283",
      "0.192546549"
    ],
    [
      "1.066942717",
      "0.192546549"
    ],
    [
      "1.200828151",
      "0.192546549"
    ],
    [
      "0.449703156",
      "0.246495337"
    ],
    [
      "1.550296844",
      "0.246495337"
    ],
    [
      "0.59839828",
      "0.304672137"
    ],
    [
      "1.40160172",
      "0.304672137"
    ],
    [
      "0.732229133",
      "0.308494736"
    ],
    [
      "0.866114566",
      "0.308494736"
    ],
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    ],
    [
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      "0.308494736"
    ],
    [
      "1.267770867",
      "0.308494736"
    ],
    [
      "0.406271757",
      "0.373140601"
    ],
    [
      "1.593728243",
      "0.373140601"
    ],
    [
      "0.27745694",
      "0.409637891"
    ],
    [
      "1.72254306",
      "0.409637891"
    ],
    [
      "0.531455563",
      "0.420620323"
    ],
    [
      "1.468544437",
      "0.420620323"
    ],
    [
      "0.665286416",
      "0.424442923"
    ],
    [
      "0.799171849",
      "0.424442923"
    ],
    [
      "0.933057283",
      "0.424442923"
    ],
    [
      "1.066942717",
      "0.424442923"
    ],
    [
      "1.200828151",
      "0.424442923"
    ],
    [
      "1.334713584",
      "0.424442923"
    ],
    [
      "0.194795964",
      "0.532485703"
    ],
    [
      "0.330689679",
      "0.532485703"
    ],
    [
      "1.669310321",
      "0.532485703"
    ],
    [
      "1.805204036",
      "0.532485703"
    ],
    [
      "0.464512846",
      "0.53656851"
    ],
    [
      "1.535487154",
      "0.53656851"
    ],
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      "0.598343699",
      "0.54039111"
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      "0.732229133",
      "0.54039111"
    ],
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      "0.54039111"
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    ],
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      "0.54039111"
    ],
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      "0.54039111"
    ],
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      "0.54039111"
    ],
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      "0.648433889"
    ],
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      "0.648433889"
    ],
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      "0.39757013",
      "0.652516697"
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      "1.60242987",
      "0.652516697"
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      "0.531400982",
      "0.656339296"
    ],
    [
      "0.665286416",
      "0.656339296"
    ],
    [
      "0.799171849",
      "0.656339296"
    ],
    [
      "0.933057283",
      "0.656339296"
    ],
    [
      "1.066942717",
      "0.656339296"
    ],
    [
      "1.200828151",
      "0.656339296"
    ],
    [
      "1.334713584",
      "0.656339296"
    ],
    [
      "1.468599018",
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  "radius": "0.066942716"
}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。 · 验证器 v1.0.0

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